Chow theory of toric variety bundles

TL;DR

This paper provides a comprehensive description of the Chow homology and cohomology of toric variety bundles, extending existing theories to include singular fibers and establishing new algebraic and combinatorial tools.

Contribution

It introduces a unified framework for understanding the Chow groups of toric bundles, relating them to Minkowski weights and piecewise polynomial functions, with applications in enumerative geometry.

Findings

Describes Chow homology and cohomology as modules over the base's cohomology.

Identifies cohomology with Minkowski weights and piecewise polynomial functions.

Establishes a fan displacement rule for Minkowski weights and limits via equivariant multiplicities.

Abstract

We describe the Chow homology and cohomology of toric variety bundles, with no restrictions on the singularities of the fibre. We present the ordinary and equivariant homologies as modules over the cohomology of the base, identify the ordinary cohomology with homology-valued Minkowski weights, and identify the equivariant cohomology with cohomology-weighted piecewise polynomial functions. We describe the product structure on Minkowski weights via a fan displacement rule, and the non-equivariant limit via equivariant multiplicities. Along the way we establish relative analogues of the K\"unneth property and Kronecker duality. Applications include the balancing condition in logarithmic enumerative geometry.